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On exact correlation functions of chiral ring operators in $2d$ $\mathcal{N}=(2, 2)$ SCFTs via localization
High Energy Physics - Theory · Submitted 4 Dec 2017
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim With the extended algorithm and technique of localization, we compute exactly the extremal correlators in $2d$ $\mathcal{N}=(2, 2)$ (twisted) chiral rings as non-holomorphic functions of marginal parameters of the theories.
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We study the extremal correlation functions of (twisted) chiral ring operators via superlocalization in $\mathcal{N}=(2, 2)$ superconformal field theories (SCFTs) with central charge $c\geq 3$, especially for SCFTs with Calabi-Yau geometric phases. We extend the method in arXiv:1602.05971 with mild modifications, so that it is applicable to disentangle operators mixing on $S^2$ in nilpotent (twisted) chiral rings of $2d$ SCFTs. With the extended algorithm and technique of localization, we compute exactly the extremal correlators in $2d$ $\mathcal{N}=(2, 2)$ (twisted) chiral rings as non-holomorphic functions of marginal parameters of the theories. Especially in the context of Calabi-Yau geometries, we give an explicit geometric interpretation to our algorithm as the Griffiths transversality with projection on the Hodge bundle over Calabi-Yau complex moduli. We also apply the method to compute extremal correlators in K\"{a}hler moduli, or say twisted chiral rings, of several interesting Calabi-Yau manifolds. In the case of complete intersections in toric varieties, we provide an alternative formalism for extremal correlators via localization onto Higgs branch. In addition, as a spinoff we find that, from the extremal correlators of the top element in twisted chiral rings, one can extract chiral correlators in A-twisted topological theories.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We study the extremal correlation functions of (twisted) chiral ring operators via superlocalization in $\mathcal{N}=(2, 2)$ superconformal field theories (SCFTs) with central charge $c\geq 3$, especially for SCFTs with Calabi-Yau geometric phases. | contribution 0.97 | 0.92 | 0.15 | asserted 0.95 |
| s1 | We extend the method in arXiv:1602.05971 with mild modifications, so that it is applicable to disentangle operators mixing on $S^2$ in nilpotent (twisted) chiral rings of $2d$ SCFTs. | method 0.59 | 0.90 | 0.08 | asserted 0.99 |
| s2 | With the extended algorithm and technique of localization, we compute exactly the extremal correlators in $2d$ $\mathcal{N}=(2, 2)$ (twisted) chiral rings as non-holomorphic functions of marginal parameters of the theories. | result 0.89 | 0.96 | 0.63 | asserted 1.00 |
| s3 | Especially in the context of Calabi-Yau geometries, we give an explicit geometric interpretation to our algorithm as the Griffiths transversality with projection on the Hodge bundle over Calabi-Yau complex moduli. | interpretation 0.98 | 0.93 | 0.15 | asserted 1.00 |
| s4 | We also apply the method to compute extremal correlators in K\"{a}hler moduli, or say twisted chiral rings, of several interesting Calabi-Yau manifolds. | contribution 0.74 | 0.95 | 0.08 | asserted 1.00 |
| s5 | In the case of complete intersections in toric varieties, we provide an alternative formalism for extremal correlators via localization onto Higgs branch. | contribution 0.71 | 0.94 | 0.07 | asserted 1.00 |
| s6 | In addition, as a spinoff we find that, from the extremal correlators of the top element in twisted chiral rings, one can extract chiral correlators in A-twisted topological theories. | result 0.95 | 0.95 | 0.10 | asserted 1.00 |
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Original abstract
We study the extremal correlation functions of (twisted) chiral ring operators via superlocalization in $\mathcal{N}=(2, 2)$ superconformal field theories (SCFTs) with central charge $c\geq 3$, especially for SCFTs with Calabi-Yau geometric phases. We extend the method in arXiv:1602.05971 with mild modifications, so that it is applicable to disentangle operators mixing on $S^2$ in nilpotent (twisted) chiral rings of $2d$ SCFTs. With the extended algorithm and technique of localization, we compute exactly the extremal correlators in $2d$ $\mathcal{N}=(2, 2)$ (twisted) chiral rings as non-holomorphic functions of marginal parameters of the theories. Especially in the context of Calabi-Yau geometries, we give an explicit geometric interpretation to our algorithm as the Griffiths transversality with projection on the Hodge bundle over Calabi-Yau complex moduli. We also apply the method to compute extremal correlators in K\"{a}hler moduli, or say twisted chiral rings, of several interesting Calabi-Yau manifolds. In the case of complete intersections in toric varieties, we provide an alternative formalism for extremal correlators via localization onto Higgs branch. In addition, as a spinoff we find that, from the extremal correlators of the top element in twisted chiral rings, one can extract chiral correlators in A-twisted topological theories.
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References
How the literature uses it
6 arXiv papers cite this in 9 sentences
Papers citing it · 6 sentences without a confident label
Compares with it 1 paper
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Note: After this paper was finished the paper [ref] came out with which this paper has a large overlap.
2D BPS Rings from Sphere Partition Functions · Prologue · 0.96
Cites it as background 2 papers
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By the “$2d/4d$ correspondence" we mean here the cases in which either some of physics contents are exactly the same, e.g. the Alday-Gaiotto-Tachikawa (AGT) correspondence [ref] , or the dynamical behaviors in 2d and 4d coincide, for instance, the BPS spectra, certain correlation functions, dualities, etc., are identical [ref] .
Comments on the NSVZ $\beta$ Functions in Two-dimensional $\mathcal N=(0,2)$ Supersymmetric Models · Introduction and conclusion · 0.86
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In this paper we limit ourselves to a detailed analysis of the $=2$ supersymmetric rank-1 theories.[Certain 2d and 3d supersymmetric theories also admit some similar classes of protected operators and extremal correlation functions can be defined [ref] .
Extremal Correlators and Random Matrix Theory · Introduction · 0.86
What it cites · 72 sentences without a confident label
Uses the method or tool of 3 papers
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Solving $z_1$, $z_2$ in terms of $t_1$ and $t_2$, one arrives at the partition function up to one-instanton correction,[Our normalization is different from [ref] by a factor of $-1/8π^3$.] Z_GN%̄s̄/̄%̄s̄103 ξ_1^-3+10 ξ_1^-2ξ_2^-1+8 ξ_1^-1ξ_2^-2+4/3 ξ_2^-3+128 ζ(3) 112 q_1+384 q_1q_2+(56 q_1+192 q_1q_2)ξ_1^-1+192 q_1q_2 ξ_2^-1+c.c.+O(q^2_1, q^2_2) , where as before we set ξ_i≡1/4π Im t_i , and q_i≡e^2πi t_i , for i=1, 2 .From $Z_GN$, we can compute all correlators on $S^2$, M_a_αb̅_β=1/Z_GN∏_i,j̅(-1/2π∂/∂t^i)^n_i(1/2π∂/∂t̅^ j̅)^n^'_j̅Z_GN .
Two-Sphere Partition Functions and Gromov-Witten Invariants · The determinantal Gulliksen-Negård CY-threefold · 0.80
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With $q=0$ and the standard method in [ref] , one can rewrite eq.
Exact Kahler Potential for Calabi-Yau Fourfolds · The complete intersection in Grassmannian: $X_1^8⊂G(2,8)$ · 1.00
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All discussions on chiral primaries equally work for twisted chiral primaries, and vice versa.] $g_IJ̅$ of SCFTs with UV Lagrangian descriptions in the fashion of [ref] .
Correlation Functions of Coulomb Branch Operators · Chiral ring data via superlocalization on $S^2$ · 0.92
Compares with 2 papers
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Finally it is interesting to compute the $4$-point chiral correlator ([ref]) in A-twisted topological theory on $S^2$, by use of Eq. ([ref]), C^(4) C^(4)|⟨σ·σ·σ·σ⟩_S^2|^2=g^(α) Z_A^2 3̄6|1 + 20160 q + 689472000 q^2 + 24691154100480 q^3 903369974818590720 q^4+O(q^5)|^2Therefore we have C^(4)(q)6̄(1 + 20160 q + 689472000 q^2 + 24691154100480 q^3 + 903369974818590720 q^4+O(q^5) ) , which perfectly matches the results in [ref] .
Mirror Manifolds in Higher Dimension · Calabi-Yau phase · 0.97
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Since the chiral correlators are determined up to a Kähler transformation, one can verify that, accompanied by a transformation [equation] C^(3)_ijk(τ̃_1,τ̃_2)⟶T(τ̃_1,τ̃_2) C^(3)_ijk(τ̃_1,τ̃_2) reproduce the result of [ref] to one-instanton order.
The equivariant A-twist and gauged linear sigma models on the two-sphere · One-instanton correction to twisted chiral ring data · 0.99
Cites as background 19 papers
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On the other hand, these correlators are known to satisfy the four-dimensional version of $tt^*$-equations [ref] .
Topological Anti-Topological Fusion in Four-Dimensional Superconformal Field Theories · Introduction · 0.86
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Our second example is the PAX/PAXY gauged linear sigma models introduced in [ref] to describe the determinantal Calabi-Yau varieties.
Nonabelian 2D Gauge Theories for Determinantal Calabi-Yau Varieties · The determinantal Gulliksen-Negård CY-threefold · 0.81
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Recently, a series of papers [ref] initiated a systematic study on the correlation functions of operators in chiral rings of four-dimensional superconformal field theories with exactly marginal couplings.
$tt^*$ equations, localization and exact chiral rings in 4d N=2 SCFTs · Introduction · 0.94
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